Optimal. Leaf size=355 \[ -\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{d-c^2 d x^2}}-\frac{x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 d}-\frac{3 b x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}-\frac{3 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 d}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt{d-c^2 d x^2}}-\frac{b^2 x^3 (1-c x) (c x+1)}{32 c^2 \sqrt{d-c^2 d x^2}}-\frac{15 b^2 x (1-c x) (c x+1)}{64 c^4 \sqrt{d-c^2 d x^2}}+\frac{15 b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{64 c^5 \sqrt{d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.02443, antiderivative size = 371, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {5798, 5759, 5676, 5662, 90, 52, 100, 12} \[ -\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{d-c^2 d x^2}}-\frac{x^3 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{3 b x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}-\frac{3 x (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt{d-c^2 d x^2}}-\frac{b^2 x^3 (1-c x) (c x+1)}{32 c^2 \sqrt{d-c^2 d x^2}}-\frac{15 b^2 x (1-c x) (c x+1)}{64 c^4 \sqrt{d-c^2 d x^2}}+\frac{15 b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{64 c^5 \sqrt{d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5798
Rule 5759
Rule 5676
Rule 5662
Rule 90
Rule 52
Rule 100
Rule 12
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{d-c^2 d x^2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{\sqrt{d-c^2 d x^2}}\\ &=-\frac{x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 c^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{2 c \sqrt{d-c^2 d x^2}}\\ &=-\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{d-c^2 d x^2}}-\frac{3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 \sqrt{d-c^2 d x^2}}+\frac{\left (3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{4 c^3 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt{d-c^2 d x^2}}-\frac{3 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}-\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{d-c^2 d x^2}}-\frac{3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 c^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 c^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{3 b^2 x (1-c x) (1+c x)}{16 c^4 \sqrt{d-c^2 d x^2}}-\frac{b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt{d-c^2 d x^2}}-\frac{3 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}-\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{d-c^2 d x^2}}-\frac{3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c^4 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{32 c^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{15 b^2 x (1-c x) (1+c x)}{64 c^4 \sqrt{d-c^2 d x^2}}-\frac{b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt{d-c^2 d x^2}}+\frac{3 b^2 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{16 c^5 \sqrt{d-c^2 d x^2}}-\frac{3 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}-\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{d-c^2 d x^2}}-\frac{3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{64 c^4 \sqrt{d-c^2 d x^2}}\\ &=-\frac{15 b^2 x (1-c x) (1+c x)}{64 c^4 \sqrt{d-c^2 d x^2}}-\frac{b^2 x^3 (1-c x) (1+c x)}{32 c^2 \sqrt{d-c^2 d x^2}}+\frac{15 b^2 \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{64 c^5 \sqrt{d-c^2 d x^2}}-\frac{3 b x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c^3 \sqrt{d-c^2 d x^2}}-\frac{b x^4 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{8 c \sqrt{d-c^2 d x^2}}-\frac{3 x (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{8 c^4 \sqrt{d-c^2 d x^2}}-\frac{x^3 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{4 c^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{8 b c^5 \sqrt{d-c^2 d x^2}}\\ \end{align*}
Mathematica [A] time = 1.52373, size = 295, normalized size = 0.83 \[ \frac{32 a^2 c \sqrt{d} x \left (c^2 x^2-1\right ) \left (2 c^2 x^2+3\right )-96 a^2 \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-4 a b \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (16 \cosh \left (2 \cosh ^{-1}(c x)\right )+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \left (6 \cosh ^{-1}(c x)+8 \sinh \left (2 \cosh ^{-1}(c x)\right )+\sinh \left (4 \cosh ^{-1}(c x)\right )\right )\right )+b^2 \sqrt{d} \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (32 \cosh ^{-1}(c x)^3-4 \left (16 \cosh \left (2 \cosh ^{-1}(c x)\right )+\cosh \left (4 \cosh ^{-1}(c x)\right )\right ) \cosh ^{-1}(c x)+8 \cosh ^{-1}(c x)^2 \left (8 \sinh \left (2 \cosh ^{-1}(c x)\right )+\sinh \left (4 \cosh ^{-1}(c x)\right )\right )+32 \sinh \left (2 \cosh ^{-1}(c x)\right )+\sinh \left (4 \cosh ^{-1}(c x)\right )\right )}{256 c^5 \sqrt{d} \sqrt{d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.48, size = 887, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (b^{2} x^{4} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname{arcosh}\left (c x\right ) + a^{2} x^{4}\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{2} d x^{2} - d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2} x^{4}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]